Question about Ideals.

Suppose that R is a commutative ring and let I be an ideal of R. Suppose
that are such that there are positive integers with .
I basically need to know if it's true that , if not, give a counter example.

My working:

Given the definition of Ideals, then

I don't think the statement is true, since the definition isn't an if and only if implication, i.e it isn't necessarily true that given letting

Can someone please help me think of a counter example, or show me that i'm wrong. Thank you.

Suppose that R is a commutative ring and let I be an ideal of R. Suppose
that are such that there are positive integers with .
I basically need to know if it's true that , if not, give a counter example.

My working:

Given the definition of Ideals, then

I don't think the statement is true, since the definition isn't an if and only if implication, i.e it isn't necessarily true that given letting

Can someone please help me think of a counter example, or show me that i'm wrong. Thank you.

The set of all multiples of 4 is an ideal in the ring of integers. It contains but it does not contain 2.